" My neighbor has two very large tower poplar planted about two feet from
the
fence line. This weedy tree sends shoots up forty feet into my yard. How
do I stop them? Also I want to plant a tree or lilac close by but I'm
afraid that if I water and feed it, the poplar will be the one to benefit.
What would be an aggressive tree to plant in the vicinity that might choke
the poplar shoots?

you need to communicate with your neighbour about this whole thing. note i
said 'communicate' - as opposed to 'confront.'